# Homework Help: Group theory

1. Jan 20, 2008

### ehrenfest

[SOLVED] group theory

1. The problem statement, all variables and given/known data
Let $\phi:G \to G'$ be a group homomorphism. Show that if |G| is finite, then $|\phi(G)|$ is finite and is a divisor of |G|.

2. Relevant equations

3. The attempt at a solution
Should the last word be |G'|? Then it would follow from Lagrange's Theorem.

2. Jan 20, 2008

### Hurkyl

Staff Emeritus
Nope; it's right as stated. (And can also use Lagrange's theorem in its proof)

3. Jan 20, 2008

### quasar987

Think first isomorphism theorem.

4. Jan 20, 2008

### ehrenfest

I haven't gotten to the first isomorphism theorem yet, but I don't even need it:

We know that $\phi^{-1}(\phi(a)) = aH = Ha$, where H = Ker(phi). So, the cardinality of phi(G) will be the index of H in G, which must divide |G| by Lagrange's Theorem.

Is that right?

5. Jan 20, 2008

Bingo.